Calculus of Variations and Geometric Measure Theory
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L. Rizzi

A counterexample to gluing theorems for MCP metric measure spaces

created by rizzi1 on 15 Nov 2017
modified on 23 Jul 2018


Published Paper

Inserted: 15 nov 2017
Last Updated: 23 jul 2018

Journal: Bulletin of the London Mathematical Society
Year: 2017
Doi: 10.1112/blms.12186

ArXiv: 1711.04499 PDF


Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq \kappa$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the $\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane, which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.

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